First, I think I would say that native speakers do not learn most words by memorizing roots. The most common vocabulary is acquired by rote memorization in the first 12 years or so of life. Furthermore, about 26% of words in English are Germanic in origin, with a greater percentage among common words, and these roots are rarely taught, with most courses focusing on Latin or Greek roots.
That being said, when learning specialized and technical (read: Greek-derived and Latin-derived) vocabulary later in life, most native speakers do learn the roots. Many test preparation guides recommend some knowledge of roots for standarized tests such as the GRE or SAT, where testers can benefit from making educated guesses. Learning roots can help one remember words (again, most often of the technical variety). Example: many educated native speakers know that the word analgesia comprises the roots an-, meaning not or none, and algesia, from the Greek Ἄλγος, meaning pain. Nearly every native speaker knows that the root phobia refers to fear. If I asked a native speaker what algophobia meant, they would stand a good chance of guessing that it referred to fear of pain. Even if they did not, once they understood the meaning, it would be quite easy to associate it with analgesia in their mind.
So roots are useful, but most native speakers did not use them to learn their base vocabulary. Refer and defer share a root, but few could tell you what it means. (It comes from the Latin ferre, meaning to carry).
Short answer: The cuts are visible marks to what he or she is feeling inside.
The above usage seems more literal than some others.
My best guess is that to put a punctuation mark on something means
to bring to (visible) completion
or
to (visibly) complete
A sentence is not complete until a punctuation mark is "put on it".
A couple other uses (from Vocabulary dot com):
Evan Berry put a punctuation mark on the blowout by returning one of Tennessee’s four interceptions 100 yards for a TD in the closing seconds.
Notice this is talking about an exceptional running play that produced more points in the blowout game, "in the closing seconds".
The use you found regarding the basketball team is similar, since it was the last home game of their season.
It used to be the family dinner was a given; the punctuation mark at the end of each day, absolute and non-negotiable.
Notice at the end of each day. So we're in the realm of something that completes something.
Wagner's final shot of the match, a grunting effortful bouncer that clipped Ian Bell, was a worthy punctuation mark to a grand effort.
NB his final shot
The riot was symptomatic of the area’s sense of isolation and served as a punctuation mark on 10 years of restiveness in Miami.
All my examples come from the same vocabulary dot com link; I had to scroll through the examples. Some refer to 'literal' punctuation marks.
As far as the example with cutting,
It expresses emotional pain or feelings that I’m unable to put into words. It puts a punctuation mark on what I’m feeling on the inside!
Could it (cutting oneself) be thought of as somehow completing what the person is "feeling on the inside"?That's not too satisfactory. However, if you consider punctuation mark in the more literal sense, as visible marks, then this is perhaps what this usage is doing. The cuts themselves are visible marks on what he or she is feeling inside.
So, I'll just add this, to put a punctuation mark on my answer:
It’s now official what many people had predicted: former Alaska Governor Sarah Palin has ended her political tease — which helped her promote books and raise money for her political groups — and has formally announced that she is not going to run for the 2012 Republican nomination:
...
The decision from Ms. Palin on Wednesday placed a punctuation mark on the Republican presidential field. Her intentions were the biggest remaining question in the race. [Emphases mine]
(The Moderate Voice, quoting The New York Times).
Best Answer
The following definition from the McGraw-Hill Dictionary may help:
off to the races: